1. Field of the Invention
This invention relates generally to the training of prefilters used to process signals received over a communications channel and, more particularly, to the training of prefilters in the frequency domain.
2. Description of the Related Art
There is an ever-increasing demand for higher communications data rates, including in wireless systems such as cellular systems and wireless local area networks. However, inter-symbol interference (ISI) is one effect that can significantly limit the performance of communications systems. In wireless systems, ISI typically is the result of “multipath delay spread,” which occurs due to the existence of multiple propagation paths with different delays between the transmitter and the receiver. In a communications channel with multipath delay spread, a single impulse transmitted from the transmitter does not arrive at the receiver as a single impulse. Rather, it arrives as a series of impulses with different attenuations and delays, with each impulse corresponding to a different propagation path. The resulting received signal impulse can be significantly longer than that of the original transmitted signal. If the communications channel is characterized by an impulse response h, an h with a longer delay spread typically causes more ISI. Effects other than multipath can also increase the delay spread of h relative to the ideal situation. For example, the presence of transmit filters (e.g., for spectral shaping) and/or receive filtering (e.g., anti-alias/channel select filtering) can also increase the delay spread. Whatever the cause, longer h can limit the performance of the receiver.
For example, the maximum likelihood sequence estimator (MLSE) is known to be the optimum (in some sense) sequence estimator for a communications channel with ISI. However, the complexity of the MLSE, as measured by the number of states in the MLSE trellis, grows exponentially with the length of the delay spread. Specifically, the number of states in the MLSE is given by MK where M is the size of the modulation alphabets and K is the delay spread of h measured in symbol durations. For communications channels that have a long delay spread, the complexity quickly grows to make the MLSE an impractical alternative. For example, the increased complexity can adversely impact the overall power budget, which is a concern for mobile applications, and/or the price, which is a concern for consumer devices.
One alternative to the MLSE is non-trellis based equalizers, such as the decision feedback equalizer (DFE). These types of equalizers typically do not suffer from the complexity of an MLSE and can produce near-optimum performance for uncoded transmission. However, one drawback of DFEs and other types of non-trellis based equalizers is that they typically generate a hard output, i.e., a final decision that the transmitted bit is either a 0 or a 1. They typically cannot generate a confidence level or probability measure that the transmitted bit is either a 0 or a 1. The latter typed output is sometimes referred to as a soft decision. In systems that use forward error correction (FEC) or other types of encoding, the equalizer or sequence estimator is followed by a decoder. Better decoding performance is achieved when soft decisions are provided to the decoder. Trellis based sequence estimators are believed to be the only known methods for generating reliable soft decisions for decoding. As a result, the use of non-trellis based equalizers with decoders is less than optimal and the problem of reducing MLSE complexity remains an important one.
Prefiltering can reduce the complexity of the MLSE. In this approach, the communications channel is followed by a prefilter in the receiver. Thus, the “effective” communications channel seen by the rest of the receiver is a combination of the communications channel with the prefilter. This combination shall be referred to as the conditioned channel. The prefilter is designed to re-shape or condition the communications channel so that the impulse response of the conditioned channel has a delay spread that is shorter than that of the original communications channel. Since the sequence estimator operates over the conditioned channel rather than over the original communications channel, the complexity of the sequence estimator is reduced accordingly.
The problem of designing (or training) a prefilter typically is based on the equationw*h≈b  (1)where h is the impulse response of the communications channel, w is the impulse response of the prefilter, b is the ideal impulse response of the conditioned channel, and * denotes linear convolution. Training a prefilter based on Eqn. 1 can be divided into two tasks. The first task is to determine the criterion for the desired conditioned channel b. The second task is to actually solve Eqn. 1 with low complexity for b and w. Low complexity is especially important since for many applications, the communications channel varies over time and the prefilter is trained on an on-going basis to account for varying conditions. For example, in mobile applications where data is transmitted in packets, quick adaptation is desired to compensate for changes due to changing position and packet-to-packet variations in the communications channel.
In one approach, constraints are imposed on the nature of w and b, and then w and b are jointly optimized in the context of Eqn. 1. However, this approach is generally complex since Eqn. 1 is based on a convolution and w typically is long in order to achieve good performance. For example, gradient-based approaches typically converge too slowly to be useful. Other methods for prefilter training rely on efficient algorithms for solving linear equations, such as the RLS algorithm or Cholesky factorization. However, these methods still require on the order of L2 or L3 computations, where L is the length of b plus the length of w. For the long prefilter lengths that are typically necessary to obtain good performance, these methods are also usually too complex.
Thus, there is a need for improved prefiltering methods with low training complexity, particularly approaches that can provide good performance over communications channels with ISI.